Prerequisite concepts
Understanding both difference (how far apart two values are at one time) and change (how far apart the value of a single parameter is at two different times) is necessary for understanding derivatives.
One can view $\frac{df}{dx}$ as approximately given by a fraction where $df$ is a small change in $f$ and $dx$ is a small change in $x$.
The value of the derivative at a point on a curve can be approximated by the slope of a line secant with the curve near that point.
Technically, the derivative is a ratio of small changes in the limit that the change in the denominator goes to zero: $\frac{df}{dx}=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$
One of Zandieh's process-object layers for derivatives is that derivatives are functions. This means that the value of a derivative depends on where in the domain of the function one is looking. The derivative of a function is itself a function, with the same domain as the original function. Both the (derivative) function and the value of the derivative at a point are commonly referred to as "the derivative."
When taking a derivative of a multivariable function, it is possible to consider a simpler situation in which some of the variables are considered to be constant.
Zapping with d
We would like our students to have both a practical and mathematical understanding of a total differential. In particular, with this activity students learn what the consequences of zapping'' a function with the operatord'' are. This activity also opens up a conversation on infinitesimals, and their relation to total differentials. The use of variables not associated with the Partial Derivative Machine in this activity allows students to apply the mathematical techniques they have been practicing during the course to a less familiar physical system.
Representations used
Concepts taught
One typical notation used for derivatives is Leibniz notation ($e.g.,$ $df/dx$). Physicists often think of the $df$ and $dx$ as distinct quantities (``differentials'') that can be measured, calculated, and manipulated independent of one another. In particular, a differential can be thought of as the (small) change in a quantity when a small change is made to a physical system. A differential can also be thought of as the difference between the values of a quantity between two different (nearby) physical states. This line of thinking is also valuable for thinking about integration in physical situations.
FIXME: Does the difference between zapping with d and taking the total differential need to be articulated?
FIXME: What? One can do algebra with differentials and reinterpret them as partial derivatives. Can treat like a variable
No matter how complicated a multivariable function is when written as an equation, the total differential of that function will be linear in the differential terms. In other words: $dF(x,y,z)=A dx + B dy + C dz.$